> Consider the set of all positive rational numbers as existing.> Fill into the vase all numbers of the interval (0, 1]. Then enumerate> one of them by 1 and take it off the vase.> Fill into the vase all numbers of the interval (1, 2]. Then enumerate> one of them by 2 and take it off the vase.> Fill into the vase all numbers of the interval (2, 3]. Then enumerate> one of them by 3 and take it off the vase.> Continue until all rational numbers have been enumerated. Then all> have been taken off the vase. The remaining set of not enumerated> rationals is empty.

> This result is nonsense from the mathematical standpoint.

You are right, your result is nonsense. If the rational numbertaken off the interval (n-1, n] is n then you define just theidentity map on N and not a map Q,>0 -> N.